Andoni-Krauthgamer-Razenshteyn对可勾画的规范的刻画对于可勾画的度量是失败的
The Andoni--Krauthgamer--Razenshteyn characterization of sketchable norms fails for sketchable metrics
摘要：
Andoni、Krauthgamer和Razenshteyn（AKR）证明（STOC 2015）有限维赋范空间$(X,\|\cdot\|_X)$允许$O(1)$草图算法（即，具有$O(1)$草图大小和$O(1)$近似）当且仅当对于每个$\varepsilon\in (0,1)$存在$α\geqslant 1$和嵌入$f:X\to \ell_{1-\varepsilon}$，使得对于所有$x,y\in X$都有$\|x-y\|_X\leqslant \|f(x)-f(y)\|_{1-\varepsilon}\leqslant α\|x-y\|_X$。这个定理的“如果部分”来自Indyk的草图算法（FOCS 2000）。因此，AKR的贡献是证明，仅仅是草图算法的可用性就意味着上述几何实现的存在。Indyk的算法表明，AKR特征的“如果部分”对任何度量空间都成立，即，即使当$X$不是赋范空间时，上述嵌入的存在也意味着可绘制性。因此，AKR提出的一个自然问题是，是否需要假设底层空间是赋范空间来表征可草绘性。我们解决这个问题证明，对于任意大的$n\in \mathbb{N}$有一个$n$ -点度量空间$(M(n),d_{M(n)})$是$O(1)$ -sketchable尚未为每个$\varepsilon\in (0,\frac12)$，如果$α(n)\geqslant 1$和$f_n:M(n)\to \ell_{1-\varepsilon}$是这样的，$d_{M(n)}(x,y)\leqslant \|f_n(x)-f_n(y)\|_{1-\varepsilon}\leqslant α(n) d_{M(n)}(x,y)$为所有$x,y\in M(n)$，那么必然$\lim_{n\to \infty} α(n)= \infty$。
英文摘要：
Andoni, Krauthgamer and Razenshteyn (AKR) proved (STOC 2015) that a finite-dimensional normed space $(X,\|\cdot\|_X)$ admits a $O(1)$ sketching algorithm (namely, with $O(1)$ sketch size and $O(1)$ approximation) if and only if for every $\varepsilon\in (0,1)$ there exist $α\geqslant 1$ and an embedding $f:X\to \ell_{1-\varepsilon}$ such that $\|x-y\|_X\leqslant \|f(x)-f(y)\|_{1-\varepsilon}\leqslant α\|x-y\|_X$ for all $x,y\in X$. The "if part" of this theorem follows from a sketching algorithm of Indyk (FOCS 2000). The contribution of AKR is therefore to demonstrate that the mere availability of a sketching algorithm implies the existence of the aforementioned geometric realization. Indyk's algorithm shows that the "if part" of the AKR characterization holds true for any metric space whatsoever, i.e., the existence of an embedding as above implies sketchability even when $X$ is not a normed space. Due to this, a natural question that AKR posed was whether the assumption that the underlying space is a normed space is needed for their characterization of sketchability. We resolve this question by proving that for arbitrarily large $n\in \mathbb{N}$ there is an $n$-point metric space $(M(n),d_{M(n)})$ which is $O(1)$-sketchable yet for every $\varepsilon\in (0,\frac12)$, if $α(n)\geqslant 1$ and $f_n:M(n)\to \ell_{1-\varepsilon}$ are such that $d_{M(n)}(x,y)\leqslant \|f_n(x)-f_n(y)\|_{1-\varepsilon}\leqslant α(n) d_{M(n)}(x,y)$ for all $x,y\in M(n)$, then necessarily $\lim_{n\to \infty} α(n)= \infty$.
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